Connected semitopological group

Definition

A semitopological group is termed connected if its underlying topological space is a connected space.

Facts

• For any semitopological group, there is a closed normal subgroup called the connected component of identity. This is the largest connected subgroup containing the identity element. It is a topological automorphism-invarant subgroup.
• Connected implies no proper open subgroup
• No proper open subgroup not implies connected
• Locally connected and no proper open subgroup implies connected